Can every coninuous signal be modelled as a function, which then can be converted into a series of sine and consine functions with unique frequencies?

And let us say that we have some arbitrary sequence of natural numbers. e.g. 1, 2, 7, 3, 17, 19. Is it possible to convert every finite and infinite sequence into some continuous function model, such as in Fourier theory?

I know that it is possible to extract some discrete samples from a continous signal/function and construct the original continuous signal, as provided by Nyquist-Shannon sampling theorem. The question is whether it is possible to construct a continuous signal that models a set of discrete samples. Can this only be approximate?